Classical Transport, Steady States and Large Deviations in Non-equilibrium 1d Systems

The goal of these three lectures was to review a series of results obtained over the last fifteen years on non-equilibrium diffusive systems [7; 8; 17; 24]. They started by an introduction to the different types of models which are studied to describe non-equilibrium systems (models with deterministic thermostats [35; 39; 52] and deterministic dynamics, models with stochastic thermostats and deterministic internal dynamics, models with stochastic thermostats [27; 47; 57; 58] and stochastic internal dynamics [40; 41; 48; 49; 56]): in the particular case of stochastic thermostats it was shown how the idea of detailed balanced can be extended to describe systems in contact with several heat baths at unequal temperatures or several reservoirs of particles at unequal chemical potentials [24; 54; 59]. Close to equilibrium there is an Einstein relation between the heat current generated by a small temperature difference between two heat baths and the variance of the heat exchanges between the two thermostats [43]. This relation can be viewed as a consequence of the fluctuation theorem [30-33; 44; 45; 50] which was discussed in the case of stochastic dynamics [45]. Finally the last part of the first lecture gave several examples which do or do not satisfy Fourier's law or Fick's law [27; 47]. The second lecture was devoted to the fluctuations and the large deviation functions of the current in non-equilibrium diffusive systems [1; 5; 6; 11-13; 25; 26; 36; 42; 51]. When the dynamics is described by a Markov process, the Legendre transform of the large deviation function of the current can be computed as the largest eigenvalue of a matrix obtained by deforming the Markov matrix [23; 24]. For arbitrary diffusive systems of large linear size the whole distribution of the fluctuations of the current can be obtained using the macroscopic fluctuation theory. Conditionned on the current, one can calculate the density profile along the system [11; 29; 38]. In some cases, this optimal profile undergoes phase transitions, for example by becoming time dependent [12; 37]. For diffusive systems at equilibrium one can also show, using the macroscopic fluctuation theory, that the distribution of the fluctuations of the current is universal [2; 46]. The third lecture was focused on a number of results about the fluctuations and the large deviation function of the density in non-equilibrium steady states. These results were obtained either by exact microscopic calculations based on the matrix ansatz [10; 18; 19] for some specific models such as the symmetric exclusion process [20; 21; 28] or using fluctuating hydrodynamics and the macroscopic fluctuation theory [3; 4; 9]. These different approaches show that diffusive systems generically exhibit long range correlations [14; 15; 22; 34; 53; 55] and a large deviation function of the density which is non local. How to generalize these results on current and density fluctuations to mechanical models which do not satisfy Fourier's law is one of the open questions raised in the conclusion [16; 57; 58].